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}{TEXT 924 0 "" }}{PARA 212 "" 0 "" {TEXT 927 89 " The notion of a slope field and Euler's Method f or approximating solutions to DE's are " }{TEXT 927 0 "" }}{PARA 213 " " 0 "" {TEXT 928 66 " described and then visualized in an example tha t utilizes some " }{TEXT 929 6 "Maple " }{TEXT 928 23 " system-based \+ commands." }{TEXT 928 0 "" }}{PARA 214 "" 0 "" {TEXT 930 0 "" }}{PARA 215 "" 0 "" {TEXT 931 53 " Recall that if we are given a differential equation" }{TEXT 931 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 47 " \+ " }{XPPEDIT 18 0 "d/dx*y:" "6#*(%\"dG\"\"\"%#dxG!\"\"%\"yGF%" } {TEXT 924 11 " = " }{TEXT 932 2 "F " }{TEXT 924 2 "( " }{TEXT 932 1 "x" }{TEXT 924 3 " , " }{TEXT 932 1 "y" }{TEXT 924 2 " )" } {TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 17 " and any point " }{XPPEDIT 18 0 "P[o];" "6#&%\"PG6#%\" oG" }{TEXT 924 8 " = ( " }{XPPEDIT 18 0 "x[o];" "6#&%\"xG6#%\"oG" } {TEXT 924 6 " , " }{XPPEDIT 18 0 "y[o];" "6#&%\"yG6#%\"oG" }{TEXT 924 27 " ) , in the domain of " }{TEXT 932 1 "F" }{TEXT 924 22 ", then we know that " }{TEXT 932 1 "F" }{TEXT 924 3 " ( " }{XPPEDIT 18 0 "x[o];" "6#&%\"xG6#%\"oG" }{TEXT 924 6 " , " }{XPPEDIT 18 0 "y [o];" "6#&%\"yG6#%\"oG" }{TEXT 924 21 " ) represents the" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 60 " slope of a solution curve when passing through the point " }{XPPEDIT 18 0 "P[o];" "6#&%\"PG6#% \"oG" }{TEXT 924 2 " " }{TEXT 932 3 " . " }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 57 " If we draw a short directed line segment h aving slope " }{TEXT 932 2 " F" }{TEXT 924 3 " ( " }{TEXT 932 1 "x" } {TEXT 924 3 " , " }{TEXT 932 1 "y" }{TEXT 924 33 " ) at a large numbe r of points " }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 2 " " } {TEXT 933 1 "P" }{TEXT 924 6 " = ( " }{TEXT 932 1 "x" }{TEXT 924 4 " \+ , " }{TEXT 932 1 "y" }{TEXT 924 34 " ) , then the result is called a " }{TEXT 934 11 "slope field" }{TEXT 935 1 " " }{TEXT 924 48 ". A slo pe field is used to visualize the general" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 67 " shapes of solution curves. Slope fields c an be generated using " }{TEXT 932 5 "Maple" }{TEXT 924 1 "." }{TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 215 "" 0 "" {TEXT 931 37 " Consider the differential equation:" }{TEXT 931 0 "" }} {PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 34 " \+ " }{TEXT 936 7 "( 1 ) " }{TEXT 924 12 " " }{XPPEDIT 18 0 "d/dx*y:" "6#*(%\"dG\"\"\"%#dxG!\"\"%\"y GF%" }{TEXT 924 14 " = " }{XPPEDIT 18 0 "exp(-x)-2*y:" "6#, &-%$expG6#,$%\"xG!\"\"\"\"\"*&\"\"#F*%\"yGF*F)" }{TEXT 924 6 " ." }{TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 25 " We will first load the " }{TEXT 932 5 "Maple" }{TEXT 924 16 " package called " }{TEXT 937 7 "DEtools" }{TEXT 924 53 ", and then plot the slope field for equ ation ( 1 ) . " }{TEXT 924 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 8 "restart:" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 14 "with(DEtools):" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 66 "DEplot (diff(y(x),x)=exp(-x)-2* y,y(x),x=-4..4,y=-4..4,color=blue);" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 923 0 "" }}}{PARA 211 "" 0 "" {TEXT 924 22 " \+ Observe the use of " }{TEXT 937 4 "diff" }{TEXT 924 8 " in the" } {TEXT 939 1 " " }{TEXT 940 6 "DEplot" }{TEXT 924 40 " above, when refe rring to the quantity " }{XPPEDIT 18 0 "dy/dx:" "6#*&%#dyG\"\"\"%#dxG !\"\"" }{TEXT 924 25 " from equation ( 1 )." }{TEXT 924 0 "" }} {PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 6 " In " }{TEXT 932 7 "Maple ," }{TEXT 924 68 " when defining an unevaluate d derivative, the standard notation is " }{TEXT 937 21 "diff( fnc(var ), var )" }{TEXT 924 2 ", " }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 7 " where" }{TEXT 939 1 " " }{TEXT 941 3 "fnc" }{TEXT 924 37 " is any unassigned function name and " }{TEXT 937 3 "var" }{TEXT 924 47 " is an unasssigned variable name. Note that the" }{TEXT 924 0 "" }} {PARA 215 "" 0 "" {TEXT 931 48 " form of the slope field plot comman d is then:" }{TEXT 931 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }} {PARA 211 "" 0 "" {TEXT 939 11 " " }{TEXT 937 80 "DEplot( [d iffeqn] , [dep var name] , [range of indep var] , [range of dep var] ) " }{TEXT 936 2 " ." }{TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 81 " The arrowheads displayed in the s lope field may be eliminated with the option " }{TEXT 937 13 "arrows \+ = line" }{TEXT 924 2 " ." }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 19 " We may even ask " }{TEXT 932 5 "Maple" }{TEXT 924 75 " to \+ plot the solution curves passing through specified initial points with " }{TEXT 924 0 "" }}{PARA 215 "" 0 "" {TEXT 931 24 " the following co mmand." }{TEXT 931 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 130 "DEplot (diff(y(x),x)=exp(-x)-2*y,y(x),x=-4..4,[ [-2,-2], [0,1], [ 0,0], [1,-1] ],y=-4..4,color=blue,linecolor=magenta,arrows=line);" } {MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 923 0 "" }}} {PARA 215 "" 0 "" {TEXT 931 1 " " }{TEXT 931 0 "" }}{PARA 211 "" 0 "" {TEXT 924 67 " Observe that the coordinates of the initial points ar e inserted " }{TEXT 932 7 "between" }{TEXT 924 15 " the range of " } {TEXT 932 1 "x" }{TEXT 924 8 " and the" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 13 " range of " }{TEXT 932 1 "y" }{TEXT 924 9 " in the " }{TEXT 941 6 "DEplot" }{TEXT 939 1 " " }{TEXT 924 8 "command." }{TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 215 "" 0 "" {TEXT 931 85 " Euler's Method uses the idea of slope fields to obtai n approximate solutions to the" }{TEXT 931 0 "" }}{PARA 215 "" 0 "" {TEXT 931 26 " initial value problem :" }{TEXT 931 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 34 " \+ " }{XPPEDIT 18 0 "dy/dx:" "6#*&%#dyG\"\"\"%#dxG! \"\"" }{TEXT 924 9 " = " }{TEXT 932 1 "F" }{TEXT 924 4 " ( " } {TEXT 932 1 "x" }{TEXT 924 3 " , " }{TEXT 932 1 "y" }{TEXT 924 14 " ) \+ , " }{TEXT 932 3 " y" }{TEXT 924 3 " ( " }{XPPEDIT 641 0 "x[ 0]:" "6#&%\"xG6#\"\"!" }{TEXT 932 11 " ) = " }{XPPEDIT 638 0 "y[ 0]:" "6#&%\"yG6#\"\"!" }{TEXT 932 2 " " }{TEXT 924 0 "" }}{PARA 210 " " 0 "" {TEXT 923 0 "" }}{PARA 215 "" 0 "" {TEXT 931 13 " as follows." }{TEXT 931 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 " " {TEXT 924 10 " Let " }{XPPEDIT 662 0 "P[0]:" "6#&%\"PG6#\"\"!" }{TEXT 932 9 " = ( " }{XPPEDIT 659 0 "x[0]:" "6#&%\"xG6#\"\"!" } {TEXT 932 6 " , " }{XPPEDIT 658 0 "y[0]:" "6#&%\"yG6#\"\"!" }{TEXT 932 10 " ) , " }{XPPEDIT 656 0 "P[1]:" "6#&%\"PG6#\"\"\"" }{TEXT 932 9 " = ( " }{XPPEDIT 654 0 "x[1]:" "6#&%\"xG6#\"\"\"" }{TEXT 932 6 " , " }{XPPEDIT 652 0 "y[1]:" "6#&%\"yG6#\"\"\"" }{TEXT 932 10 " ) , " }{XPPEDIT 649 0 "P[2]:" "6#&%\"PG6#\"\"#" }{TEXT 932 10 " = ( " }{XPPEDIT 647 0 "x[2]:" "6#&%\"xG6#\"\"#" }{TEXT 932 6 " , " }{XPPEDIT 643 0 "y[2]:" "6#&%\"yG6#\"\"#" }{TEXT 932 9 " \+ ) , " }{TEXT 942 7 ". . ." }{TEXT 932 5 " , " }{XPPEDIT 18 0 "P[ n]:" "6#&%\"PG6#%\"nG" }{TEXT 924 1 " " }{TEXT 932 9 " = ( " } {XPPEDIT 18 0 "x[n]:" "6#&%\"xG6#%\"nG" }{TEXT 924 1 " " }{TEXT 932 4 " , " }{XPPEDIT 18 0 "y[n]:" "6#&%\"yG6#%\"nG" }{TEXT 924 1 " " } {TEXT 932 5 " ) ," }{TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 25 " where " } {XPPEDIT 637 0 "x[1]:" "6#&%\"xG6#\"\"\"" }{TEXT 932 9 " = " } {XPPEDIT 635 0 "x[0]:" "6#&%\"xG6#\"\"!" }{TEXT 932 34 " + h \+ " }{XPPEDIT 633 0 "y[1]:" "6#&%\"yG6#\"\"\"" } {TEXT 932 9 " = " }{XPPEDIT 631 0 "y[0]:" "6#&%\"yG6#\"\"!" } {TEXT 932 15 " + F ( " }{XPPEDIT 629 0 "x[0]:" "6#&%\"xG6#\"\" !" }{TEXT 932 7 " , " }{XPPEDIT 627 0 "y[0]:" "6#&%\"yG6#\"\"!" } {TEXT 932 7 " ) h" }{TEXT 924 0 "" }}{PARA 215 "" 0 "" {TEXT 931 6 " " }{TEXT 931 0 "" }}{PARA 211 "" 0 "" {TEXT 924 30 " \+ " }{XPPEDIT 625 0 "x[2]:" "6#&%\"xG6#\"\"#" }{TEXT 932 8 " = " }{XPPEDIT 623 0 "x[1]:" "6#&%\"xG6#\"\"\"" }{TEXT 932 36 " + h " }{XPPEDIT 621 0 "y[2]:" "6#&%\"yG6#\"\"#" }{TEXT 932 9 " = " }{XPPEDIT 618 0 "y[1]:" "6# &%\"yG6#\"\"\"" }{TEXT 932 13 " + F ( " }{XPPEDIT 617 0 "x[1]:" "6#&%\"xG6#\"\"\"" }{TEXT 932 6 " , " }{XPPEDIT 615 0 "y[1]:" "6#&% \"yG6#\"\"\"" }{TEXT 932 7 " ) h" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 37 " " }{TEXT 936 54 ". ." }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 37 " \+ " }{TEXT 936 54 ". \+ ." }{TEXT 924 0 "" }}{PARA 217 "" 0 "" {TEXT 943 91 " \+ . \+ ." }{TEXT 943 0 "" }}{PARA 217 "" 0 "" {TEXT 943 91 " \+ . \+ ." }{TEXT 943 0 "" }}{PARA 211 "" 0 "" {TEXT 924 32 " \+ " }{XPPEDIT 18 0 "x[n]:" "6#&%\"xG6#%\"nG" } {TEXT 924 9 " = " }{XPPEDIT 612 0 "x[n-1]:" "6#&%\"xG6#,&%\"nG\" \"\"F(!\"\"" }{TEXT 932 26 " + h " }{XPPEDIT 18 0 "y[n]:" "6#&%\"yG6#%\"nG" }{TEXT 924 1 " " }{TEXT 932 8 " = " } {XPPEDIT 610 0 "y[n-1]:" "6#&%\"yG6#,&%\"nG\"\"\"F(!\"\"" }{TEXT 932 12 " + F ( " }{XPPEDIT 608 0 "x[n-1]:" "6#&%\"xG6#,&%\"nG\"\"\"F( !\"\"" }{TEXT 932 7 " , " }{XPPEDIT 607 0 "y[n-1]:" "6#&%\"yG6#,&% \"nG\"\"\"F(!\"\"" }{TEXT 932 10 " ) h ." }{TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 13 " Then le t " }{TEXT 932 1 "y" }{TEXT 924 4 " = " }{TEXT 932 1 "Q" }{TEXT 924 2 "[ " }{TEXT 932 2 "h " }{TEXT 924 43 "] denote the polygonal pa th which joins " }{XPPEDIT 606 0 "P[0]:" "6#&%\"PG6#\"\"!" }{TEXT 932 8 " to " }{XPPEDIT 605 0 "P[1]:" "6#&%\"PG6#\"\"\"" }{TEXT 932 8 " to " }{XPPEDIT 604 0 "P[2]:" "6#&%\"PG6#\"\"#" }{TEXT 932 8 " to " }{TEXT 942 8 " . . ." }{TEXT 932 6 " to " }{XPPEDIT 18 0 "P[n]:" "6#&%\"PG6#%\"nG" }{TEXT 924 1 " " }{TEXT 932 5 " ." } {TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 58 " Under fairly genera l conditions, it can be shown that " }{TEXT 932 1 "Q" }{TEXT 924 1 " [" }{TEXT 932 3 " h " }{TEXT 924 15 "] converges to " }{TEXT 932 1 "y" }{TEXT 924 2 "( " }{TEXT 932 1 "x" }{TEXT 924 7 " ) as " }{TEXT 932 2 " h" }{TEXT 924 16 " tends to zero," }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 10 " (where " }{TEXT 932 1 "y" }{TEXT 924 3 " ( " } {TEXT 932 1 "x" }{TEXT 924 34 " ) is the actual solution of the " } {TEXT 936 3 "IVP" }{TEXT 924 3 " )." }{TEXT 924 0 "" }}{PARA 215 "" 0 "" {TEXT 931 53 " We test this method on the initial value problem : " }{TEXT 931 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 19 " " }{TEXT 936 7 " ( 2 ) " }{TEXT 924 10 " " }{XPPEDIT 18 0 "dy/dx:" "6#*&%#dyG\"\"\"%#dxG!\"\" " }{TEXT 924 12 " = " }{XPPEDIT 18 0 "exp(-x)-2*y:" "6#,&-%$e xpG6#,$%\"xG!\"\"\"\"\"*&\"\"#F*%\"yGF*F)" }{TEXT 924 25 " , \+ y ( " }{XPPEDIT 18 0 "-2:" "6#,$\"\"#!\"\"" }{TEXT 924 8 " \+ ) = " }{XPPEDIT 18 0 "-2:" "6#,$\"\"#!\"\"" }{TEXT 924 4 " ." } {TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 63 " First note that the right hand side of ( 2 ) is a fu nction " }{TEXT 932 2 " F" }{TEXT 924 34 " which depends on two varia bles, " }{TEXT 932 2 " x" }{TEXT 924 6 " and " }{TEXT 932 1 "y" } {TEXT 924 2 " ." }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 17 " W e may define " }{TEXT 932 1 "F" }{TEXT 924 6 " in " }{TEXT 932 5 "M aple" }{TEXT 924 12 " as follows:" }{TEXT 924 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 22 "F:=(x,y)->exp(-x)-2*y;" }{MPLTEXT 1 938 0 "" }}}{PARA 215 "" 0 "" {TEXT 931 2 " " }{TEXT 931 0 "" }}{PARA 211 "" 0 "" {TEXT 924 61 " We will now approximate the solution of t his IVP over the " }{TEXT 932 1 "x" }{TEXT 924 11 "-interval " } {TEXT 932 1 "I" }{TEXT 924 7 " = [ " }{XPPEDIT 18 0 "-2;" "6#,$\"\"# !\"\"" }{TEXT 924 5 " , " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$\"\"#! \"\"" }{TEXT 924 5 " ]." }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 44 " We begin by using a large step size, say " }{TEXT 932 2 " h " }{TEXT 924 56 " = 0.5 , (this number represents the difference betwe en " }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 8 " the " } {TEXT 932 1 "x" }{TEXT 924 18 "-coordinates of " }{XPPEDIT 603 0 "P[ i+1]:" "6#&%\"PG6#,&%\"iG\"\"\"F(F(" }{TEXT 932 10 " and " } {XPPEDIT 18 0 "P[i]:" "6#&%\"PG6#%\"iG" }{TEXT 924 1 " " }{TEXT 932 46 " for i = 0 , 1 , 2 , . . . , n -1 ). " }{TEXT 924 0 "" }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 7 "h:=0.5;" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 9 "X[0]:=-2;" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 9 "Y[0]:=-2;" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 18 "P[0]:=[X[0],Y[0]]; " }{MPLTEXT 1 938 0 "" }}}{PARA 218 "" 0 "" {TEXT 944 2 " " }{TEXT 945 6 " Note" }{TEXT 946 39 ": Since there are 5 intervals of length" }{TEXT 945 2 " h" }{TEXT 946 20 " = 0.5 from a = " }{XPPEDIT 387 0 "-2;" "6#,$\"\"#!\"\"" }{TEXT 946 15 " to b = " }{XPPEDIT 383 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 947 2 " " }{TEXT 946 19 " , we must compute" }{TEXT 948 0 "" }}{PARA 219 "" 0 "" {TEXT 949 43 " the vertices of the points " }{XPPEDIT 377 0 "P[1] ;" "6#&%\"PG6#\"\"\"" }{TEXT 949 5 " , " }{XPPEDIT 375 0 "P[2];" "6# &%\"PG6#\"\"#" }{TEXT 949 13 " , . . . , " }{XPPEDIT 373 0 "P[5];" " 6#&%\"PG6#\"\"&" }{TEXT 949 49 " for the Euler approximation. Below , we define" }{TEXT 949 0 "" }}{PARA 219 "" 0 "" {TEXT 949 35 " \+ the function: kMax( " }{TEXT 950 1 "h" }{TEXT 949 58 " ) which determines the number of subintervals of length " }{TEXT 950 1 "h" } {TEXT 949 7 " which" }{TEXT 949 0 "" }}{PARA 220 "" 0 "" {TEXT 951 76 " occur in the partition of the interval [ a , b ] . " }{TEXT 951 0 "" }}{PARA 221 "" 0 "" {TEXT 952 6 " " } {TEXT 952 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 13 "a:=-2;b :=1/2;" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 17 "kMax:=h->(b-a)/h;" }{MPLTEXT 1 938 0 "" }}}{PARA 214 "" 0 "" {TEXT 930 0 "" }}{PARA 218 "" 0 "" {TEXT 948 4 " " }{TEXT 944 17 " \+ Note that in the" }{TEXT 953 4 " for" }{TEXT 944 22 " loop which follo ws, " }{TEXT 954 1 "k" }{TEXT 944 44 " ranges from 0 to kMax( 0.5 ) - 1 = 4." }{TEXT 948 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 32 "for k from 0 to (kMax(h) - 1) do" }{MPLTEXT 1 938 0 "" }} {PARA 216 "> " 0 "" {MPLTEXT 1 938 15 "X[k+1]:=X[k]+h:" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 28 "Y[k+1]:=Y[k]+F(X[k], Y[k])*h:" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 24 "P[k+1]:=[X[k+1],Y[k+1]]:" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 3 "od;" }{MPLTEXT 1 938 0 "" }}}{PARA 215 "" 0 "" {TEXT 931 2 " " }{TEXT 931 0 "" }}{PARA 211 "" 0 "" {TEXT 924 78 " A t this point, we may display a graph of the Euler polygonal approximat ion " }{TEXT 932 1 "Q" }{TEXT 924 8 "[0.5] ." }{TEXT 924 0 "" }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 33 "Q[0.5]:=[seq(P[k],k=0..k Max(h))];" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 29 "plot(Q[0.5],a..b,color=navy);" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 923 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 43 " Observe that when w e ask Maple to plot a " }{TEXT 937 4 "list" }{TEXT 924 56 " of ordered pairs, it automatically connects successive " }{TEXT 924 0 "" }} {PARA 215 "" 0 "" {TEXT 931 30 " points with a straight line." } {TEXT 931 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 215 "" 0 "" {TEXT 931 106 " To obtain the actual solution to the DE ( 1 ) we may \+ seperate variables, or use Maple's equation solver." }{TEXT 931 0 "" } }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 41 "dsolve(diff(y(x),x)=exp (-x)-2*y(x),y(x));" }{MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 9 " Here, " }{TEXT 955 3 "_Cl " }{TEXT 956 1 " " }{TEXT 924 87 " denotes an arbitrary constant term \+ in the solution of the differential equation ( 1 )." }{TEXT 924 0 "" } }{PARA 211 "" 0 "" {TEXT 924 2 " " }{TEXT 932 5 "Maple" }{TEXT 924 64 " will solve the related IVP ( 2 ) if the initial condition " }{TEXT 932 2 "y " }{TEXT 924 2 "( " }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\" \"" }{TEXT 924 9 " ) = " }{XPPEDIT 18 0 "-2 " "6#,$\"\"#!\"\"" } {TEXT 924 22 " is inserted in the" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 2 " " }{TEXT 937 6 "dsolve" }{TEXT 936 1 " " }{TEXT 924 20 " command as follows:" }{TEXT 924 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 52 "dsolve(\{diff(y(x),x)=exp(-x)-2*y(x),y(-2)=-2 \},y(x));" }{MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }} {PARA 215 "" 0 "" {TEXT 931 80 " We now ask for this result to be s implified. " }{TEXT 931 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 12 "simplify(%);" }{MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 215 "" 0 "" {TEXT 931 77 " Recall that to convert an expression into a function we can \+ use the command" }{TEXT 931 0 "" }}{PARA 211 "" 0 "" {TEXT 924 34 " \+ " }{TEXT 939 1 " " }{TEXT 937 1 " " } {TEXT 936 1 " " }{TEXT 924 1 "[" }{TEXT 936 1 " " }{TEXT 924 1 ">" } {TEXT 937 1 " " }{TEXT 957 3 "fcn" }{TEXT 939 1 " " }{TEXT 937 1 ":" } {TEXT 939 1 " " }{TEXT 937 9 "= unapply" }{TEXT 939 1 " " }{TEXT 937 2 "( " }{TEXT 957 4 "expr" }{TEXT 937 3 " , " }{TEXT 957 3 "var" } {TEXT 937 2 " )" }{TEXT 939 1 " " }{TEXT 937 1 ";" }{TEXT 924 0 "" }} {PARA 211 "" 0 "" {TEXT 924 26 " which assigns the name " }{TEXT 957 3 "fcn" }{TEXT 937 1 " " }{TEXT 924 33 " to the mapping of the var iable " }{TEXT 957 3 "var" }{TEXT 932 1 " " }{TEXT 924 22 " into the \+ expression " }{TEXT 957 4 "expr" }{TEXT 924 2 " ." }{TEXT 924 0 "" }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 21 "Y:=unapply(rhs(%),x);" } {MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 43 " We may plot a graph which displays both " } {TEXT 932 1 "Q" }{TEXT 924 12 "[0.5] and " }{TEXT 932 1 "Y" }{TEXT 924 2 " (" }{TEXT 932 2 " x" }{TEXT 924 4 " ) ." }{TEXT 924 0 "" }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 28 "with(plots):with(plottoo ls):" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 35 "A:=plot(Y(x),x=a..b,color=magenta):" }{MPLTEXT 1 938 0 "" }}} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 34 "B:=plot(Q[0.5],x=a..b,co lor=navy):" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 13 "display(A,B);" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 923 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" } }{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 64 " \+ We can improve the approximation by reducing the step size to " } {TEXT 932 2 "h " }{TEXT 924 6 "= 0.1." }{TEXT 924 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 7 "h:=0.1;" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 9 "X[0]:=-2;" }{MPLTEXT 1 938 0 "" }} {PARA 216 "> " 0 "" {MPLTEXT 1 938 9 "Y[0]:=-2;" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 18 "P[0]:=[X[0],Y[0]];" } {MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 222 "" 0 "" {TEXT 958 2 " " }{TEXT 959 6 " Note" }{TEXT 958 41 ": Since there are 25 intervals of length " } {TEXT 959 1 "h" }{TEXT 958 20 " = 0.1 from a = " }{XPPEDIT 328 0 " -2;" "6#,$\"\"#!\"\"" }{TEXT 959 2 " " }{TEXT 958 13 " to b = " }{XPPEDIT 325 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 960 1 " " } {TEXT 959 1 " " }{TEXT 958 19 " , we must compute" }{TEXT 960 0 "" }} {PARA 223 "" 0 "" {TEXT 961 43 " the vertices of the poi nts " }{XPPEDIT 322 0 "P[1];" "6#&%\"PG6#\"\"\"" }{TEXT 962 2 " " } {TEXT 961 3 " , " }{XPPEDIT 320 0 "P[2];" "6#&%\"PG6#\"\"#" }{TEXT 961 13 " , . . . , " }{XPPEDIT 318 0 "P[25];" "6#&%\"PG6#\"#D" } {TEXT 961 33 " for the Euler approximation. " }{TEXT 963 0 "" }} {PARA 219 "" 0 "" {TEXT 949 67 " Therefore, in the for l oop which follows, we allow " }{TEXT 950 2 "k " }{TEXT 949 43 " to ra nge from 0 to kMax( 0.1 ) -1 = 24." }{TEXT 949 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 32 "fo r k from 0 to (kMax(h) - 1) do" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 16 "X[k+1]:=X[k]+h: " }{MPLTEXT 1 938 0 "" }} {PARA 216 "> " 0 "" {MPLTEXT 1 938 28 "Y[k+1]:=Y[k]+F(X[k],Y[k])*h:" } {MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 24 "P[k+1]:=[ X[k+1],Y[k+1]]:" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 3 "od:" }{MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 33 "Q[0.1]:=[seq(P[k],k=0 ..kMax(h))]:" }{MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 38 " We now plot a graph which displays " }{TEXT 932 1 "Q" }{TEXT 924 9 "[0.5] , " }{TEXT 932 1 "Q" }{TEXT 924 13 "[0.1] and " }{TEXT 932 1 "Y" }{TEXT 924 3 " ( " }{TEXT 932 1 "x" }{TEXT 924 4 " ) ." }{TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 36 "C:=plot(Q[0.1],x=a..b,color=maroon):" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 15 "display(A, B,C);" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 " " }}{PARA 211 "" 0 "" {TEXT 924 96 " We can attain an even better app roximation to the solution curve by reducing the step size to " } {TEXT 932 2 "dx" }{TEXT 924 8 " = 0.05." }{TEXT 924 0 "" }}{PARA 210 " " 0 "" {TEXT 923 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 8 "h :=0.05;" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 9 " X[0]:=-2;" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 9 "Y[0]:=-2;" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 18 "P[0]:=[X[0],Y[0]];" }{MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 219 "" 0 "" {TEXT 949 50 " In the for loop \+ which follows, observe that " }{TEXT 950 1 "k" }{TEXT 949 63 " must n ow range from 0 to kMax( 0.05 ) - 1 = 49 in order to" }{TEXT 949 0 "" }}{PARA 222 "" 0 "" {TEXT 958 45 " obtain the Euler approxima tion over the " }{TEXT 959 2 "x " }{TEXT 958 12 "- interval " }{TEXT 959 1 "I" }{TEXT 958 7 " = [ " }{XPPEDIT 351 0 "-2;" "6#,$\"\"#!\"\" " }{TEXT 958 5 " , " }{XPPEDIT 349 0 "1/2;" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 964 2 " " }{TEXT 958 8 " ] as " }{TEXT 959 1 "h" }{TEXT 958 9 " = 0.05 ." }{TEXT 960 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" } }{EXCHG {PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 30 "for k from 0 to (kMa x(h)-1) do" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 16 "X[k+1]:=X[k]+h: " }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 28 "Y[k+1]:=Y[k]+F(X[k],Y[k])*h:" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 24 "P[k+1]:=[X[k+1],Y[k+1]]:" } {MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 3 "od:" } {MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 43 " Now plot a final graph which displays " } {TEXT 932 1 "Q" }{TEXT 924 9 "[0.5] , " }{TEXT 932 1 "Q" }{TEXT 924 8 "[0.1] , " }{TEXT 932 1 "Q" }{TEXT 924 13 "[0.05] and " }{TEXT 932 1 "Y" }{TEXT 924 3 " ( " }{TEXT 932 1 "x" }{TEXT 924 4 " ) ." } {TEXT 924 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 34 "Q[0.05] :=[seq(P[k],k=0..kMax(h))]:" }{MPLTEXT 1 938 0 "" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 938 39 "F:=plot(Q[0.05],x=-2..0.5,color=green):" }{MPLTEXT 1 938 0 "" }}{PARA 216 "> " 0 "" {MPLTEXT 1 938 17 "display( A,B,C,F);" }{MPLTEXT 1 938 0 "" }}}{PARA 210 "" 0 "" {TEXT 923 0 "" }} {PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 224 "" 0 "" {TEXT 965 1 " " } {TEXT 966 154 " \+ \+ " }{TEXT 965 2 " " }{TEXT 967 5 " " }{TEXT 967 0 "" }}{PARA 211 "" 0 "" {TEXT 924 2 " " }{TEXT 936 9 "Exe rcises" }{TEXT 924 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 924 2 " " }{TEXT 936 3 "1. " }{TEXT 924 31 " For th e differential equation:" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 38 " " }{XPPEDIT 18 0 "dy/dx; " "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT 924 8 " + 2" }{TEXT 932 2 "y \+ " }{TEXT 924 15 " = 1 + sin 4" }{TEXT 932 1 "x" }{TEXT 924 2 " " } {XPPEDIT 18 0 "-2;" "6#,$\"\"#!\"\"" }{TEXT 924 8 " cos 6" }{TEXT 932 1 "x" }{TEXT 924 0 "" }}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 968 23 "restart: with(student):" }{MPLTEXT 1 968 0 "" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 968 0 "" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 968 58 "dsolve(\{diff(y(x),x)=-2*y(x)+1+sin(4*x)-2*cos(6*x)\},y(x)); " }{MPLTEXT 1 968 0 "" }}}{EXCHG {PARA 225 "> " 0 "" {MPLTEXT 1 968 0 "" }}}{PARA 211 "" 0 "" {TEXT 924 6 " " }{TEXT 936 7 " ( a )" } {TEXT 924 52 " Plot a direction field over the plane region: 0 " } {TEXT 969 1 "<" }{TEXT 924 2 " " }{TEXT 932 1 "x" }{TEXT 924 2 " " } {TEXT 969 1 "<" }{TEXT 924 9 " 10 , " }{XPPEDIT 18 0 "-1:" "6#,$\" \"\"!\"\"" }{TEXT 924 5 " " }{TEXT 969 1 "<" }{TEXT 924 2 " " } {TEXT 932 1 "y" }{TEXT 924 2 " " }{TEXT 969 1 "<" }{TEXT 924 6 " 3 . " }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 5 " " }{TEXT 936 8 " ( b )" }{TEXT 924 38 " To see more detail, add the option " } {TEXT 937 22 "dirgrid = [ 50 , 50 ] " }{TEXT 924 10 " to your " } {TEXT 937 6 "DEplot" }{TEXT 924 10 " command " }{TEXT 924 0 "" }} {PARA 215 "" 0 "" {TEXT 931 35 " from part ( a ) ." } {TEXT 931 0 "" }}{PARA 211 "" 0 "" {TEXT 924 7 " " }{TEXT 936 8 " ( c ) " }{TEXT 924 7 "Use a " }{TEXT 932 5 "Maple" }{TEXT 924 52 " command to add solution curves through the points:" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 36 " ( 0 , \+ " }{XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT 924 58 " ) , ( 0 , \+ 0 ) , ( 0 , 1 ) , ( 0 , 2 ) and ( 0 , 3 )" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 60 " to the direction field. Th is time, use the" }{TEXT 936 1 " " }{TEXT 924 7 "option " }{TEXT 937 25 " dirgrid = [ 100 , 100 ] " }{TEXT 924 9 " in your " }{TEXT 924 0 " " }}{PARA 211 "" 0 "" {TEXT 924 19 " " }{TEXT 937 6 "DEplot" }{TEXT 924 10 " command." }{TEXT 924 0 "" }}{PARA 211 "" 0 " " {TEXT 924 7 " " }{TEXT 936 8 " ( d ) " }{TEXT 924 85 " From y our graph in part ( c ) , what appears to happen to all of the solutio n curves" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 21 " \+ as " }{TEXT 932 1 "x" }{TEXT 924 15 " becomes large?" }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 9 " " }{TEXT 936 7 "( e ) " }{TEXT 924 5 "Use " }{TEXT 932 5 "Maple" }{TEXT 924 77 " to so lve the differential equation. Then explain how this general solution \+ " }{TEXT 924 0 "" }}{PARA 215 "" 0 "" {TEXT 931 57 " \+ supports your assertion in part ( d ) ." }{TEXT 931 0 "" }}{PARA 210 " " 0 "" {TEXT 923 0 "" }}{PARA 211 "" 0 "" {TEXT 936 6 " 2. " }{TEXT 924 4 " " }{XPPEDIT 18 0 "dy/dx:" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT 924 8 " = " }{XPPEDIT 18 0 "4*x-2*y/x:" "6#,&*&\"\"%\"\"\"% \"xGF&F&*(\"\"#F&%\"yGF&F'!\"\"F+" }{TEXT 924 3 " " }{TEXT 936 1 ";" }{TEXT 924 17 " initial point" }{TEXT 936 2 " :" }{TEXT 924 2 " " }{XPPEDIT 18 0 "P[o]" "6#&%\"PG6#%\"oG" }{TEXT 924 7 " = " } {TEXT 970 1 "(" }{TEXT 924 2 " " }{XPPEDIT 18 0 "1/2:" "6#*&\"\"\"F$ \"\"#!\"\"" }{TEXT 924 8 " , 3 " }{TEXT 970 1 ")" }{TEXT 924 2 " " }{TEXT 936 1 ";" }{TEXT 924 7 " " }{TEXT 932 1 "x" }{TEXT 924 12 " - interval " }{TEXT 936 2 ": " }{TEXT 924 2 " " }{TEXT 932 1 "I" }{TEXT 924 5 " = " }{TEXT 970 1 "[" }{TEXT 924 1 " " }{XPPEDIT 18 0 "1/2:" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 924 10 " , + 2 " }{TEXT 970 1 "]" }{TEXT 924 2 " ." }{TEXT 924 0 "" }}{PARA 215 "" 0 "" {TEXT 931 1 " " }{TEXT 931 0 "" }}{PARA 211 "" 0 "" {TEXT 936 7 " ( a )" } {TEXT 924 55 " Plot a direction field for the DE over the region: " }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" }{TEXT 924 3 " " }{TEXT 969 1 "<" }{TEXT 924 2 " " }{TEXT 932 1 "x" }{TEXT 924 2 " " }{TEXT 969 1 "<" }{TEXT 924 8 " 2 , " }{XPPEDIT 18 0 "-4:" "6#,$\"\"%!\"\"" } {TEXT 924 5 " " }{TEXT 969 1 "<" }{TEXT 924 2 " " }{TEXT 932 1 "y " }{TEXT 924 2 " " }{TEXT 969 1 "<" }{TEXT 924 7 " 8 . " }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 936 8 " ( b ) " }{TEXT 924 61 " Ad d some solution curves to the slope field from part ( a ) " }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 936 7 " ( c )" }{TEXT 924 91 " F ind the solution to the DE by hand. Then check to determine if the app roximate solution" }{TEXT 924 0 "" }}{PARA 215 "" 0 "" {TEXT 931 65 " \+ curves plotted in part ( b ) have appropriate shapes. " } {TEXT 931 0 "" }}{PARA 211 "" 0 "" {TEXT 936 8 " ( d ) " }{TEXT 924 6 " Use " }{TEXT 932 5 "Maple" }{TEXT 924 38 " to solve the differen tial equation ." }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 2 " " }{TEXT 936 7 "( e ) " }{TEXT 924 6 " Use " }{TEXT 932 5 "Maple" } {TEXT 924 77 " to find the particular solution to the DE which passes through the point " }{XPPEDIT 18 0 "P[o]" "6#&%\"PG6#%\"oG" }{TEXT 924 4 " ." }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 2 " " } {TEXT 936 5 "( f )" }{TEXT 924 34 " Find the Euler approximations " }{TEXT 932 1 "Q" }{TEXT 924 8 "[0.5] , " }{TEXT 932 1 "Q" }{TEXT 924 13 "[0.1] , and " }{TEXT 932 1 "Q" }{TEXT 924 39 "[0.05] to the sol ution curve passing " }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 26 " through point " }{XPPEDIT 18 0 "P[o];" "6#&%\"PG6#%\"o G" }{TEXT 924 23 " over the interval " }{TEXT 932 3 "I =" }{TEXT 924 10 " [ a , b ]" }{TEXT 932 2 " ." }{TEXT 924 0 "" }}{PARA 211 "" 0 "" {TEXT 924 2 " " }{TEXT 936 6 "( g ) " }{TEXT 924 100 " Plot the \+ polygonal approximations together with the actual solution curve obta ined in part ( e ) " }{TEXT 924 0 "" }}{PARA 215 "" 0 "" {TEXT 931 41 " on a single coordinate plane." }{TEXT 931 0 "" }}}{PARA 207 "" 0 "" {TEXT 920 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 210 "" 0 "" {TEXT 923 0 "" }}{PARA 226 "" 0 "" {TEXT 971 0 "" }}{PARA 227 "" 0 "" {TEXT 972 0 "" }}{PARA 228 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }